Worked example 1
Integrate a sine function
Find the integral of 6 sin(3x + 1) with respect to x.
- Use the sine integration rule.
- Divide 6 by the inner coefficient 3.
- Include the negative cosine and + C.
Answer: -2 cos(3x + 1) + C
Higher Mathematics
Higher Maths Trigonometric integration
Practise trigonometric integration for Scottish Higher Mathematics with worked examples, clear methods and original interactive questions. Integrate p sin(qx + r) and p cos(qx + r), keeping the sign and inner factor correct.
Before you start
- Know trig derivatives.
- Work in radians.
- Track negative signs.
Explanation
Trigonometric integration reverses the derivatives of sine and cosine. The inner coefficient is removed by division.
Because the derivative of cosine is negative sine, integrating sine introduces a negative sign.
Method and rules
- Integral of p cos(qx + r) dx = sin(qx + r) + C
- Integral of p sin(qx + r) dx = − cos(qx + r) + C
- Differentiate to verify the sign and multiplier.
Worked examples
Worked example 2
Method check
Integrate 8 cos(4x − 3)
- Divide the outside coefficient by 4.
So: 2 sin(4x − 3) + C
Watch out
- Missing the negative sign when integrating sine.
- Multiplying by q instead of dividing by q.
- Treating the angle as degrees.
Exam reminder
For trigonometric integration, show the defining equation or formula before simplifying. Check that every final value satisfies the original restrictions, interval or context.
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